In India, I guess Hyderabad would be a good option. Also Mumbai is good, as there are lots of theme parks around and offcourse the beaches. You could try Goa as well if ur kids are little older. take them to Sahakari Spice Farm and you always can spend time on the beaches there (South Goan beaches). Outside I feel Singapore and Dubai are two very good options. There are loads of theme parks as well as other kids friendly options. Vegan food wouldn't be a problem in both of these places.
Places like hampi , Ooty , mysuru in karnataka ,mahableshwar in maharashtra are among the kids friendly places . You can also visit Dandeli it is considered as adventure hub of karnataka where all age group people can enjoy to the fullest and make the vacation memorable